mathlib documentation


Lean mathlib notes

Various implementation details are noted in the mathlib source, and referenced later on. We collect these notes here.

Design choices about smooth algebraic structures

  1. All smooth algebraic structures on G are Prop-valued classes that extend smooth_manifold_with_corners I G. This way we save users from adding both [smooth_manifold_with_corners I G] and [has_smooth_mul I G] to the assumptions. While many API lemmas hold true without the smooth_manifold_with_corners I G assumption, we're not aware of a mathematically interesting monoid on a topological manifold such that (a) the space is not a smooth_manifold_with_corners; (b) the multiplication is smooth at (a, b) in the charts ext_chart_at I a, ext_chart_at I b, ext_chart_at I (a * b).

  2. Because of model_prod we can't assume, e.g., that a lie_group is modelled on π“˜(π•œ, E). So, we formulate the definitions and lemmas for any model.

  3. While smoothness of an operation implies its continuity, lemmas like has_continuous_mul_of_smooth can't be instances becausen otherwise Lean would have to search for has_smooth_mul I G with unknown π•œ, E, H, and I : model_with_corners π•œ E H. If users needs [has_continuous_mul G] in a proof about a smooth monoid, then they need to either add [has_continuous_mul G] as an assumption (worse) or use haveI in the proof (better).

referenced by: [1] [2] [3] [4] [5] [6]

addition on function coercions

Terms containing @has_add.add (has_coe_to_fun.F ...) pi.has_add seem to cause leanchecker to crash due to an out-of-memory condition. As a workaround, we add a type annotation: (f + g : V₁ β†’ Vβ‚‚)

bundled maps over different rings

Frequently, we find ourselves wanting to express a bilinear map M β†’β‚—[R] N β†’β‚—[R] P or an equivalence between maps (M β†’β‚—[R] N) ≃ₗ[R] (M' β†’β‚—[R] N') where the maps have an associated ring R. Unfortunately, using definitions like these requires that R satisfy comm_semiring R, and not just semiring R. Using M β†’β‚—[R] N β†’+ P and (M β†’β‚—[R] N) ≃+ (M' β†’β‚—[R] N') avoids this problem, but throws away structure that is useful for when we do have a commutative (semi)ring.

To avoid making this compromise, we instead state these definitions as M β†’β‚—[R] N β†’β‚—[S] P or (M β†’β‚—[R] N) ≃ₗ[S] (M' β†’β‚—[R] N') and require smul_comm_class S R on the appropriate modules. When the caller has comm_semiring R, they can set S = R and smul_comm_class_self will populate the instance. If the caller only has semiring R they can still set either R = β„• or S = β„•, and add_comm_monoid.nat_smul_comm_class or add_comm_monoid.nat_smul_comm_class' will populate the typeclass, which is still sufficient to recover a ≃+ or β†’+ structure.

An example of where this is used is linear_map.prod_equiv.

referenced by: [1] [2] [3] [4] [5] [6] [7]

category_theory universes

The typeclass category C describes morphisms associated to objects of type C : Type u.

The universe levels of the objects and morphisms are independent, and will often need to be specified explicitly, as category.{v} C.

Typically any concrete example will either be a small_category, where v = u, which can be introduced as

universes u
variables {C : Type u} [small_category C]

or a large_category, where u = v+1, which can be introduced as

universes u
variables {C : Type (u+1)} [large_category C]

In order for the library to handle these cases uniformly, we generally work with the unconstrained category.{v u}, for which objects live in Type u and morphisms live in Type v.

Because the universe parameter u for the objects can be inferred from C when we write category C, while the universe parameter v for the morphisms can not be automatically inferred, through the category theory library we introduce universe parameters with morphism levels listed first, as in

universes v u


universes v₁ vβ‚‚ u₁ uβ‚‚

when multiple independent universes are needed.

This has the effect that we can simply write category.{v} C (that is, only specifying a single parameter) while u will be inferred.

Often, however, it's not even necessary to include the .{v}. (Although it was in earlier versions of Lean.) If it is omitted a "free" universe will be used.

classical lemma

We make decidability results that depends on classical.choice noncomputable lemmas.

coercion into rings

Coercions such as nat.cast_coe that go from a concrete structure such as β„• to an arbitrary ring Ξ± should be set up as follows:

@[priority 900] instance : has_coe_t β„• Ξ± := ⟨...⟩

It needs to be has_coe_t instead of has_coe because otherwise type-class inference would loop when constructing the transitive coercion β„• β†’ β„• β†’ β„• β†’ .... The reduced priority is necessary so that it doesn't conflict with instances such as has_coe_t Ξ± (option Ξ±).

For this to work, we reduce the priority of the coe_base and coe_trans instances because we want the instances for has_coe_t to be tried in the following order:

  1. has_coe_t instances declared in mathlib (such as has_coe_t Ξ± (with_top Ξ±), etc.)
  2. coe_base, which contains instances such as has_coe (fin n) n
  3. nat.cast_coe : has_coe_t β„• Ξ± etc.
  4. coe_trans

If coe_trans is tried first, then nat.cast_coe doesn't get a chance to apply.

referenced by: [1]

custom simps projection

You can specify custom projections for the @[simps] attribute. To do this for the projection my_structure.original_projection by adding a declaration my_structure.simps.my_projection that is definitionally equal to my_structure.original_projection but has the projection in the desired (simp-normal) form. Then you can call

initialize_simps_projections (original_projection β†’ my_projection, ...)

to register this projection.

Running initialize_simps_projections without arguments is not necessary, it has the same effect if you just add @[simps] to a declaration.

If you do anything to change the default projections, make sure to call either @[simps] or initialize_simps_projections in the same file as the structure declaration. Otherwise, you might have a file that imports the structure, but not your custom projections.

referenced by: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

decidable arguments

As mathlib is primarily classical, if the type signature of a def or lemma does not require any decidable instances to state, it is preferable not to introduce any decidable instances that are needed in the proof as arguments, but rather to use the classical tactic as needed.

In the other direction, when decidable instances do appear in the type signature, it is better to use explicitly introduced ones rather than allowing Lean to automatically infer classical ones, as these may cause instance mismatch errors later.

decidable namespace

In most of mathlib, we use the law of excluded middle (LEM) and the axiom of choice (AC) freely. The decidable namespace contains versions of lemmas from the root namespace that explicitly attempt to avoid the axiom of choice, usually by adding decidability assumptions on the inputs.

You can check if a lemma uses the axiom of choice by using #print axioms foo and seeing if classical.choice appears in the list.

dsimp, simp

Many proofs in the category theory library use the dsimp, simp pattern, which typically isn't necessary elsewhere.

One would usually hope that the same effect could be achieved simply with simp.

The essential issue is that composition of morphisms involves dependent types. When you have a chain of morphisms being composed, say f : X ⟢ Y and g : Y ⟢ Z, then simp can operate succesfully on the morphisms (e.g. if f is the identity it can strip that off).

However if we have an equality of objects, say Y = Y', then simp can't operate because it would break the typing of the composition operations. We rarely have interesting equalities of objects (because that would be "evil" --- anything interesting should be expressed as an isomorphism and tracked explicitly), except of course that we have plenty of definitional equalities of objects.

dsimp can apply these safely, even inside a composition.

After dsimp has cleared up the object level, simp can resume work on the morphism level --- but without the dsimp step, because simp looks at expressions syntactically, the relevant lemmas might not fire.

There's no bound on how many times you potentially could have to switch back and forth, if the simp introduced new objects we again need to dsimp. In practice this does occur, but only rarely, because simp tends to shorten chains of compositions (i.e. not introduce new objects at all).

referenced by: [1]

forgetful inheritance

Suppose that one can put two mathematical structures on a type, a rich one R and a poor one P, and that one can deduce the poor structure from the rich structure through a map F (called a forgetful functor) (think R = metric_space and P = topological_space). A possible implementation would be to have a type class rich containing a field R, a type class poor containing a field P, and an instance from rich to poor. However, this creates diamond problems, and a better approach is to let rich extend poor and have a field saying that F R = P.

To illustrate this, consider the pair metric_space / topological_space. Consider the topology on a product of two metric spaces. With the first approach, it could be obtained by going first from each metric space to its topology, and then taking the product topology. But it could also be obtained by considering the product metric space (with its sup distance) and then the topology coming from this distance. These would be the same topology, but not definitionally, which means that from the point of view of Lean's kernel, there would be two different topological_space instances on the product. This is not compatible with the way instances are designed and used: there should be at most one instance of a kind on each type. This approach has created an instance diamond that does not commute definitionally.

The second approach solves this issue. Now, a metric space contains both a distance, a topology, and a proof that the topology coincides with the one coming from the distance. When one defines the product of two metric spaces, one uses the sup distance and the product topology, and one has to give the proof that the sup distance induces the product topology. Following both sides of the instance diamond then gives rise (definitionally) to the product topology on the product space.

Another approach would be to have the rich type class take the poor type class as an instance parameter. It would solve the diamond problem, but it would lead to a blow up of the number of type classes one would need to declare to work with complicated classes, say a real inner product space, and would create exponential complexity when working with products of such complicated spaces, that are avoided by bundling things carefully as above.

Note that this description of this specific case of the product of metric spaces is oversimplified compared to mathlib, as there is an intermediate typeclass between metric_space and topological_space called uniform_space. The above scheme is used at both levels, embedding a topology in the uniform space structure, and a uniform structure in the metric space structure.

Note also that, when P is a proposition, there is no such issue as any two proofs of P are definitionally equivalent in Lean.

To avoid boilerplate, there are some designs that can automatically fill the poor fields when creating a rich structure if one doesn't want to do something special about them. For instance, in the definition of metric spaces, default tactics fill the uniform space fields if they are not given explicitly. One can also have a helper function creating the rich structure from a structure with less fields, where the helper function fills the remaining fields. See for instance uniform_space.of_core or real_inner_product.of_core.

For more details on this question, called the forgetful inheritance pattern, see Competing inheritance paths in dependent type theory: a case study in functional analysis.

referenced by: [1] [2] [3] [4] [5] [6] [7]

function coercion

Many structures such as bundled morphisms coerce to functions so that you can transparently apply them to arguments. For example, if e : Ξ± ≃ Ξ² and a : Ξ± then you can write e a and this is elaborated as ⇑e a. This type of coercion is implemented using the has_coe_to_fun type class. There is one important consideration:

If a type coerces to another type which in turn coerces to a function, then it must implement has_coe_to_fun directly:

structure sparkling_equiv (Ξ± Ξ²) extends Ξ± ≃ Ξ²

-- if we add a `has_coe` instance,
instance {Ξ± Ξ²} : has_coe (sparkling_equiv Ξ± Ξ²) (Ξ± ≃ Ξ²) :=

-- then a `has_coe_to_fun` instance **must** be added as well:
instance {Ξ± Ξ²} : has_coe_to_fun (sparkling_equiv Ξ± Ξ²) :=
⟨λ _, Ξ± β†’ Ξ², Ξ» f, f.to_equiv.to_fun⟩

(Rationale: if we do not declare the direct coercion, then ⇑e a is not in simp-normal form. The lemma coe_fn_coe_base will unfold it to ⇑↑e a. This often causes loops in the simplifier.)

referenced by: [1] [2] [3] [4] [5]

implicit instance arguments

There are places where typeclass arguments are specified with implicit {} brackets instead of the usual [] brackets. This is done when the instances can be inferred because they are implicit arguments to the type of one of the other arguments. When they can be inferred from these other arguments, it is faster to use this method than to use type class inference.

For example, when writing lemmas about (f : Ξ± β†’+* Ξ²), it is faster to specify the fact that Ξ± and Ξ² are semirings as {rΞ± : semiring Ξ±} {rΞ² : semiring Ξ²} rather than the usual [semiring Ξ±] [semiring Ξ²].

referenced by: [1] [2] [3] [4] [5] [6]

is_R_or_C instance

This instance generates a type-class problem with a metavariable ?m that should satisfy is_R_or_C ?m. Since this can only be satisfied by ℝ or β„‚, this does not cause problems.

likely generated binder names

In surface Lean, we can write anonymous Ξ  binders (i.e. binders where the argument is not named) using the function arrow notation:

inductive test : Type
| intro : unit β†’ test

After elaboration, however, every binder must have a name, so Lean generates one. In the example, the binder in the type of intro is anonymous, so Lean gives it the name αΎ°:

test.intro : βˆ€ (αΎ° : unit), test

When there are multiple anonymous binders, they are named αΎ°_1, αΎ°_2 etc.

Thus, when we want to know whether the user named a binder, we can check whether the name follows this scheme. Note, however, that this is not reliable. When the user writes (for whatever reason)

inductive test : Type
| intro : βˆ€ (αΎ° : unit), test

we cannot tell that the binder was, in fact, named.

The function name.is_likely_generated_binder_name checks if a name is of the form αΎ°, αΎ°_1, etc.

referenced by: [1] [2]

lower instance priority

Certain instances always apply during type-class resolution. For example, the instance add_comm_group.to_add_group {Ξ±} [add_comm_group Ξ±] : add_group Ξ± applies to all type-class resolution problems of the form add_group _, and type-class inference will then do an exhaustive search to find a commutative group. These instances take a long time to fail. Other instances will only apply if the goal has a certain shape. For example int.add_group : add_group β„€ or {Ξ± Ξ²} [add_group Ξ±] [add_group Ξ²] : add_group (Ξ± Γ— Ξ²). Usually these instances will fail quickly, and when they apply, they are almost the desired instance. For this reason, we want the instances of the second type (that only apply in specific cases) to always have higher priority than the instances of the first type (that always apply). See also #1561.

Therefore, if we create an instance that always applies, we set the priority of these instances to 100 (or something similar, which is below the default value of 1000).

referenced by: [1]

module definition

Modules are defined as an abbreviation for semimodules, if the base semiring is a ring. (A previous definition made module a structure defined to be semimodule.) This has as advantage that modules are completely transparent for type class inference, which means that all instances for semimodules are immediately picked up for modules as well. A cosmetic disadvantage is that one can not extend modules as such, in definitions such as normed_space. The solution is to extend semimodule instead.

no instance on morphisms

We have lemmas stating that the composition of two morphisms is again a morphism. Since composition is reducible, type class inference will always succeed in applying these instances. For example when the goal is just ⊒ is_mul_hom f the instance is_mul_hom.comp will still succeed, unifying f with f ∘ (λ x, x). This causes type class inference to loop. To avoid this, we do not make these lemmas instances.


Currently, the linter forbids the use of > and β‰₯ in definitions and statements, as they cause problems in rewrites. They are still allowed in statements such as bounded (β‰₯) or βˆ€ Ξ΅ > 0 or ⨆ n β‰₯ m, and the linter allows that. If you write a pattern where you bind two or more variables, like βˆƒ n m > 0, the linter will flag this as illegal, but it is also allowed. In this case, add the line

@[nolint ge_or_gt] -- see Note [nolint_ge]
referenced by: [1]

open expressions

Some declarations work with open expressions, i.e. an expr that has free variables. Terms will free variables are not well-typed, and one should not use them in tactics like infer_type or unify. You can still do syntactic analysis/manipulation on them. The reason for working with open types is for performance: instantiating variables requires iterating through the expression. In one performance test pi_binders was more than 6x quicker than mk_local_pis (when applied to the type of all imported declarations 100x).

referenced by: [1] [2] [3] [4] [5] [6] [7]

operator precedence of big operators

There is no established mathematical convention for the operator precedence of big operators like ∏ and βˆ‘. We will have to make a choice.

Online discussions, such as seem to suggest that ∏ and βˆ‘ should have the same precedence, and that this should be somewhere between * and +. The latter have precedence levels 70 and 65 respectively, and we therefore choose the level 67.

In practice, this means that parentheses should be placed as follows:

βˆ‘ k in K, (a k + b k) = βˆ‘ k in K, a k + βˆ‘ k in K, b k β†’
  ∏ k in K, a k * b k = (∏ k in K, a k) * (∏ k in K, b k)

(Example taken from page 490 of Knuth's Concrete Mathematics.)

partially-applied ext lemmas

When possible, ext lemmas are stated without a full set of arguments. As an example, for bundled homs f, g, and of, f.comp of = g.comp of β†’ f = g is a better ext lemma than (βˆ€ x, f (of x) = g (of x)) β†’ f = g, as the former allows a second type-specific extensionality lemmas to be applied to f.comp of = g.comp of. If the domain of of is β„• or β„€ and of is a ring_hom, such a lemma could then make the goal f (of 1) = g (of 1).

For bundled morphisms, there is a ext lemma that always applies of the form (βˆ€ x, ⇑f x = ⇑g x) β†’ f = g. When adding type-specific ext lemmas like the one above, we want these to be tried first. This happens automatically since the type-specific lemmas are inevitably defined later.

referenced by: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

simp-normal form

This note gives you some tips to debug any errors that the simp-normal form linter raises.

The reason that a lemma was considered faulty is because its left-hand side is not in simp-normal form. These lemmas are hence never used by the simplifier.

This linter gives you a list of other simp lemmas: look at them!

Here are some tips depending on the error raised by the linter:

  1. 'the left-hand side reduces to XYZ': you should probably use XYZ as the left-hand side.

  2. 'simp can prove this': This typically means that lemma is a duplicate, or is shadowed by another lemma:

    2a. Always put more general lemmas after specific ones:

    @[simp] lemma zero_add_zero : 0 + 0 = 0 := rfl
    @[simp] lemma add_zero : x + 0 = x := rfl

    And not the other way around! The simplifier always picks the last matching lemma.

    2b. You can also use @[priority] instead of moving simp-lemmas around in the file.

    Tip: the default priority is 1000. Use @[priority 1100] instead of moving a lemma down, and @[priority 900] instead of moving a lemma up.

    2c. Conditional simp lemmas are tried last. If they are shadowed just remove the simp attribute.

    2d. If two lemmas are duplicates, the linter will complain about the first one. Try to fix the second one instead! (You can find it among the other simp lemmas the linter prints out!)

  3. 'try_for tactic failed, timeout': This typically means that there is a loop of simp lemmas. Try to apply squeeze_simp to the right-hand side (removing this lemma from the simp set) to see what lemmas might be causing the loop.

    Another trick is to set_option trace.simplify.rewrite true and then apply try_for 10000 { simp } to the right-hand side. You will see a periodic sequence of lemma applications in the trace message.

use has_coe_t

We use the class has_coe_t instead of has_coe if the first argument is a variable, or if the second argument is a variable not occurring in the first. Using has_coe would cause looping of type-class inference. See

referenced by: [1]

user attribute parameters

For performance reasons, it is inadvisable to use user_attribute.get_param. The parameter is stored as a reflected expression. When calling get_param, the stored parameter is evaluated using eval_expr, which first compiles the expression into VM bytecode. The unevaluated expression is available using user_attribute.get_param_untyped.

In particular, user_attribute.get_param MUST NEVER BE USED in the implementation of an attribute cache. This is because calling eval_expr disables the attribute cache.

There are several possible workarounds:

  1. Set a different attribute depending on the parameter.
  2. Use your own evaluation function instead of eval_expr, such as e.g. expr.to_nat.
  3. Write your own has_reflect Param instance (using a more efficient serialization format). The user_attribute code unfortunately checks whether the expression has the correct type, but you can use `(id %%e : Param) to pretend that your expression e has type Param.
referenced by: [1] [2]

vector space definition

Vector spaces are defined as an abbreviation for semimodules, if the base ring is a field. (A previous definition made vector_space a structure defined to be module.) This has as advantage that vector spaces are completely transparent for type class inference, which means that all instances for semimodules are immediately picked up for vector spaces as well. A cosmetic disadvantage is that one can not extend vector spaces as such, in definitions such as normed_space. The solution is to extend semimodule instead.